cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A354263 Expansion of e.g.f. 1/(1 + 3 * log(1-x)).

Original entry on oeis.org

1, 3, 21, 222, 3132, 55242, 1169262, 28873800, 814870584, 25871762016, 912684973968, 35416732159872, 1499286521185776, 68757945743286576, 3395829155786528976, 179693346163010491008, 10142543588881013369856, 608262031900883147262336
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Column k=3 of A320079.
Cf. A335531.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n, 3^k*k!*abs(stirling(n, k, 1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * k! * |Stirling1(n, k)|.
a(n) ~ n! * exp(n/3) / (3 * (exp(1/3) - 1)^(n+1)). - Vaclav Kotesovec, Jun 04 2022

A320080 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - k*log(1 + x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 2, 0, 1, 4, 15, 28, 4, 0, 1, 5, 28, 114, 172, 14, 0, 1, 6, 45, 296, 1152, 1328, 38, 0, 1, 7, 66, 610, 4168, 14562, 12272, 216, 0, 1, 8, 91, 1092, 11020, 73376, 220842, 132480, 600, 0, 1, 9, 120, 1778, 24084, 248870, 1550048, 3907656, 1633344, 6240, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k - 1)*x^2/2! + 2*k*(3*k^2 - 3*k + 1)*x^3/3! + 2*k*(12*k^3 - 18*k^2 + 11*k - 3)*x^4/4! + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   1,     6,     15,     28,      45,  ...
  0,   2,    28,    114,    296,     610,  ...
  0,   4,   172,   1152,   4168,   11020,  ...
  0,  14,  1328,  14562,  73376,  248870,  ...

Crossrefs

Columns k=0..5 give A000007, A006252, A088501, A335531, A354147, A365604.
Main diagonal gives A317172.
Cf. A048594, A048994, A094416, A320079, A334369.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[1/(1 - k Log[1 + x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: 1/(1 - k*log(1 + x)).
A(n,k) = Sum_{j=0..n} Stirling1(n,j)*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (-1)^(j-1) * (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A347020 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(1/3).

Original entry on oeis.org

1, 1, 3, 18, 150, 1644, 22116, 353856, 6554376, 138001896, 3254445144, 84979363248, 2433814616592, 75858381808416, 2556180134677152, 92597465283789312, 3588434497019272320, 148134619713440384640, 6489652665043455707520, 300712023388466713739520
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 11 2021

Keywords

Crossrefs

Cf. A006252, A007559, A320343, A335531, A346982, A347015, A347021, A347022, A347023.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - 3 Log[1 + x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007559(k).
a(n) ~ n! * exp(1/9) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A350720 a(n) = Sum_{k=0..n} k! * 3^k * k^n * Stirling1(n,k).

Original entry on oeis.org

1, 3, 69, 3948, 422082, 72567522, 18304992558, 6367730357160, 2921446409138136, 1709074810258369776, 1241694104839498851552, 1096850187800368469477424, 1157691464039682741551221152, 1438880771284303822650674399664
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A320083, A350719.
Cf. A195263, A335531, A350721.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * 3^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*3^k*k^n*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*log(1+k*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (3 * log(1 + k*x))^k.

A354289 Expansion of e.g.f. (1 + x)^(3/(1 - 3 * log(1+x))).

Original entry on oeis.org

1, 3, 24, 276, 4086, 73620, 1557702, 37770138, 1030916484, 31245154164, 1040274476208, 37716394860936, 1478413316987424, 62274364390387656, 2804282634867538248, 134397620584518275928, 6828489621874434752208, 366547074721109281366128
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088819, A354288.
Cf. A000262, A335531, A354287, A354291.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(3/(1-3*log(1+x)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A335531(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * A000262(k) * Stirling1(n,k).
a(n) ~ exp(-11/12 + 1/(6*(exp(1/3) - 1)) + 2*exp(1/6)*sqrt(n)/sqrt(3*(exp(1/3) - 1)) - n) * n^(n - 1/4) / (sqrt(2) * 3^(1/4) * (exp(1/3) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

A354750 Expansion of e.g.f. 1 / (1 - log(1 + 3*x) / 3).

Original entry on oeis.org

1, 1, -1, 6, -48, 534, -7542, 129240, -2603736, 60292512, -1577546928, 46021512096, -1480976147664, 52110720451152, -1990258155061776, 81995762243700864, -3624527727510038784, 171109526616468957312, -8591991935936929932672, 457246520477143117555968
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2022

Keywords

Crossrefs

Cf. A006252, A087674, A255927, A335531, A352069, A354237, A354263, A354751.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - Log[1 + 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 3^(n - k), {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+3*x)/3))) \\ Michel Marcus, Jun 06 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 3^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-3)^(k-1) * a(n-k).

A335530 Expansion of e.g.f. (1 - 2*log(1 + x))/(1 - 3*log(1 + x)).

Original entry on oeis.org

1, 1, 5, 38, 384, 4854, 73614, 1302552, 26339832, 599220000, 15146634096, 421152109344, 12774687166224, 419781904240464, 14855313525059664, 563252540698636416, 22779973705779470592, 978886224493465845888, 44538419222894143142784
Offset: 0

Views

Author

Seiichi Manyama, Jun 12 2020

Keywords

Crossrefs

Column k=3 of A334369.
Cf. A088501, A335531.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * 3^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Jun 12 2020 *)
With[{nn=20},CoefficientList[Series[(1-2Log[1+x])/(1-3Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2021 *)
  • PARI
    {a(n) = if(n==0, 1, sum(k=0, n, k!*3^(k-1)*stirling(n, k, 1)))}
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace((1-2*log(1+x))/(1-3*log(1+x))))

Formula

a(0)=1 and a(n) = Sum_{k=0..n} k! * 3^(k-1) * Stirling1(n,k) for n > 0.
a(n) ~ n! * exp(1/3) / (9*(exp(1/3)-1)^(n+1)). - Vaclav Kotesovec, Jun 12 2020

A365599 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(2/3).

Original entry on oeis.org

1, 2, 8, 54, 498, 5868, 83940, 1413480, 27375240, 599437440, 14641665120, 394657325280, 11635613604000, 372469741813440, 12864889063033920, 476870475257550720, 18882021780125953920, 795381867831610978560, 35515223076159203880960
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2023

Keywords

Crossrefs

Cf. A335531, A347020.
Cf. A365575.

Programs

  • Mathematica
    a[n_] := Sum[Product[3*j + 2, {j, 0, k - 1}] * StirlingS1[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 3*j+2)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j+2)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 - k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/3) * n^(n + 1/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1/3) - 1)^(n + 2/3) * exp(n - 2/9)). - Vaclav Kotesovec, Nov 11 2023
Showing 1-8 of 8 results.

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